Stuck on "Waiting for participants" but can hear other people

### Steps to reproduce

Unknown

Seems the client managed to get itself into a state where it was stuck on 'waiting for participants' but it was joined to the livekit room, receiving media from other people and playing their audio. Rageshake shows, "tried to update participants before local room member is available" which implies the client has lost its own member event, presumably though state being corrupted / lost somewhere between synapse, js-sdk and js-sdk indexeddb storage.

### Outcome

#### What did you expect?

#### What happened instead?

### Operating system

_No response_

### Browser information

_No response_

### URL for webapp

_No response_

### Will you send logs?

Yes

Start by searching for the rageshake message "tried to update participants before local room member is available" and trace participant updates across the client, Matrix JS SDK, IndexedDB state, and the LiveKit room. Reproduce or construct a state where media is received while the local member is missing; done means the client no longer remains on "Waiting for participants" in that state.

Written by the indexing model from the issue text.